Implementing vector addition in Coq

问题

Implementing vector addition in some of the dependently typed languages (such as Idris) is fairly straightforward. As per the example on Wikipedia:

import Data.Vect

%default total

pairAdd : Num a => Vect n a -> Vect n a -> Vect n a
pairAdd Nil       Nil       = Nil
pairAdd (x :: xs) (y :: ys) = x + y :: pairAdd xs ys

(Note how Idris' totality checker automatically infers that addition of Nil and non-Nil vectors is a logical impossibility.)

I am trying to implement the equivalent functionality in Coq, using a custom vector implementation, albeit very similar to the one provided in the official Coq libraries:

Set Implicit Arguments.

Inductive vector (X : Type) : nat -> Type :=
  | vnul : vector X 0 
  | vcons {n : nat} (h : X) (v : vector X n) : vector X (S n).
   Arguments vnul [X].

Fixpoint vpadd {n : nat} (v1 v2 : vector nat n) : vector nat n :=
  match v1 with
  | vnul => vnul
  | vcons _ x1 v1' =>
    match v2 with
    | vnul => False_rect _ _
    | vcons _ x2 v2' => vcons (x1 + x2) (vpadd v1' v2')
    end
  end.

When Coq attempts to check vpadd, it yields the following error:

Error:
In environment
vpadd : forall n : nat, vector nat n -> vector nat n -> vector nat n
[... other types]
n0 : nat
v1' : vector nat n0
n1 : nat
v2' : vector nat n1
The term "v2'" has type "vector nat n1" while it is expected to have type "vector nat n0".

Note that, I use False_rect to specify the impossible case, otherwise the totality check wouldn't pass. However, for some reason the type checker doesn't manage to unify n0 with n1.

What am I doing wrong?

回答1:

It's not possible to implement this function so easily in plain Coq: you need to rewrite your function using the convoy pattern. There was a similar question posted a while ago about this. The idea is that you need to make your match return a function in order to propagate the relation between the indices:

Set Implicit Arguments.

Inductive vector (X : Type) : nat -> Type :=
  | vnul : vector X 0
  | vcons {n : nat} (h : X) (v : vector X n) : vector X (S n).
   Arguments vnul [X].

Definition vhd (X : Type) n (v : vector X (S n)) : X :=
  match v with
  | vcons _ h _ => h
  end.

Definition vtl (X : Type) n (v : vector X (S n)) : vector X n :=
  match v with
  | vcons _ _ tl => tl
  end.

Fixpoint vpadd {n : nat} (v1 v2 : vector nat n) : vector nat n :=
  match v1 in vector _ n return vector nat n -> vector nat n with
  | vnul =>           fun _  => vnul
  | vcons _ x1 v1' => fun v2 => vcons (x1 + vhd v2) (vpadd v1' (vtl v2))
  end v2.

来源：https://stackoverflow.com/questions/28487158/implementing-vector-addition-in-coq